W.25 Snellius–Pothenot Resection & Hansen’s Surveying Problem

Exact Algebraic Coordinate Recovery from Unmeasured Stations via Spread Triples

In Snellius-Pothenot resection, an observer P measuring spreads to three known beacons A, B, C lies on the algebraic intersection of two subtended-spread conics; in Hansen’s problem, knowing one baseline quadrance Q(A, B) and four sighting spreads at unobserved stations C, D determines every side and diagonal quadrance exactly via Triple Spread Systems and Alternate Spreads.

Interactive GEOMETOR construction for W.25 (Snellius–Pothenot Resection & Hansen’s Surveying Problem). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point D : Point ===== q_hansen = polygon(A, B, C, D) d_ac = segment(A, C) d_bd = segment(B, D)

Mathematical Synthesis

Two famous benchmarks of 17th–19th century geodesy showcase the computational power of Rational Trigonometry (Wildberger2005). In Willebrord Snellius’ resection problem (1615, later revisited by Pothenot), an observer measures spreads \(r_1 = s(P A, P B)\) and \(r_2 = s(P B, P C)\) toward three known landmarks; each spread defines an explicit circumcircle chord locus (W.17), reducing resection to circle-circle intersection.

In Peter Andreas Hansen’s two-station problem (1841), landmarks A and B have known quadrance \(Q(A, B) = 26\), while an observer at unknown stations C and D (qhansen, diagonals dac, dbd) measures only the rational spreads among lines \(C A, C B, C D\) and \(D A, D B, D C\). Combining two Triple Spread Formula relations with the Alternate Spreads ratio (W.14) recovers all four unknown quadrances (\(Q(A,D)=17, Q(B,C)=17, Q(A,C)=41, Q(B,D)=25\)) with zero rounding error!

Key Algebraic Identities

\[\frac{s(C A, C B)\, s(C B, C D)\, s(D A, A C)}{s(D A, D B)\, s(C D, C A)\, s(D B, B C)} = \frac{x}{y} \quad (\text{Spread Law Ratio Elimination})\]

\[Q(A, D) = Q(A, B)\,\frac{y}{s(D A, D B)}, \qquad Q(B, C) = Q(A, B)\,\frac{x}{s(C A, C B)}\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Known baseline quadrance Q(A, B)	26	A=[0,0], B=[1,5]
Observed rational spreads at C & D	441/697, 169/170, 361/425, 121/170	Pure rational sightings
Recovered Q(A,D), Q(B,C), Q(A,C), Q(B,D)	17, 17, 41, 25	100% exact match with zero trig!
Snellius-Pothenot resection residual at D	0	Verified == 0

Cross-Repository Canon

Executable Demonstration	W.25 Snellius–Pothenot Resection & Hansen’s Surveying Problem — Code (Standalone model.py & w-25.svg)
Thematic Part	Part IV: Practical Solvers, 3D Platonic Geometry & Polar Curves
Geodetic Canon	Willebrord Snellius (1615) & Peter Andreas Hansen (1841)
GEOMETOR Elements	G.3 extremities of a segment, G.7 plane surface
GEOMETOR Fundamentals	F.7 Ceva and Menelaus, F.12 Coaxial Circles
Exact Solver	Verified symbolically in geometor.wildberger.solvers.solve_hansen_benchmark

Constructive Dependency Graph

Depends on: W.10 Law V: Triple Spread, Triple Cross & Quadruple Spread Formulas, W.14 Menelaus, Ceva & The Alternate Spreads Theorem, W.17 Circle Geometry: Subtended Spreads, Secant Products & Rational Points, W.24 Exact Triangle Solvers & Surveying Triangulation